Math A - 3.7 Test Fall 0 Name Show your work for credit. Write all responses on separate paper. Don t use a calculator.. Open cylindrical tubes resonate at the approimate frequencies, f, where nv f = L+ 0.8d Here n is a positive integer, v is the speed of sound in air (approimately 343 meters/sec on Earth near sea level), L is the length of the tube (in meters) and d is the diameter of the resonance tube (meters). a. Find the rate of change of the frequency with respect to i. The length (when n and d and v are constant) ii. The diameter (when n and v and L are constant) iii. The velocity (when n and L and d are constant) b. For d = 0.5, nv = 400, use the differential df to approimate the change in frequency when the length of the tube increases from L = 0.9 to L =.0 Recall that Δf df = f '0.9 Δ L. The chart at right tabulates the populations (in millions) of Gabon and Gambia in the years 960 and 980. Let (t) and (t) represent the populations Gabon and Gambia, respectively. Assume that each population follows an eponential growth model: = 0 e rt, where t = 0 in the year 960. a. Set up equation by evaluating the ratio, ( 0) ( 0), in terms of both the data and the formula. Then solve these to find the relative growth rates r and r for Gabon and Gambia populations, respectively. Simplify what you can without using a calculator (don t appro. the logarithms.) b. What equation would you solve to predict, based on this evidence, the time when the populations of Gabon and Gambia are equal? Don t solve the equation here, maybe later today.
3. A camera is positioned at the top of a tower of height 0.3 km. The base of the tower is km from the base of a rocket launching pad, as shown in the diagram. Assume the height (in km) of the rocket at time t (in sec) is given by () t = 0.003t Let θ be the angle of elevation as shown in the diagram at right. a. Find an equation relating and θ in terms of t. b. Find the rate of change of θ when θ = 0. 4. The gas law for a fied mass m (in moles) of an ideal gas at absolute temperature T (in Kelvins), pressure (in atmospheres), and volume V (in liters) is V = mrt, where R 0.8 is the gas constant. Suppose that, at some instant, = 0 atm and is decreasing at a rate of 0. atm/sec, the volume is V = 8 L and V is increasing at a rate of 0.5 L/sec. Find the rate of change of the temperature at that instant if there are 00 moles of gas. 5. Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a sphere with diameter 6 cm. Hint: The paint increases the volume of sphere slightly, and the formula for the 4 3 volume V of a sphere in terms of its radius r is V = π r. 3 6. Find a formula for the derivative function for y sinh d = ( ) Recall that sinh =. + 7. At what point along the curve y = cosh() does the tangent line have slope?
Math A - 3.7 Test Solutions Fall 0.. Open cylindrical tubes resonate at the approimate frequencies, f, where nv f = ( L+ 0.8d) Here n is a positive integer, v is the speed of sound in air (approimately 343 meters/sec on Earth near sea level), L is the length of the tube (in meters) and d is the diameter of the resonance tube (meters). a. Find the rate of change of the frequency with respect to i. The length (when n and d and v are constant) d nv dl ( L+ 0.8d) ii. The diameter (when n and v and L are constant) (curious variable collision here ) d 0.4nv dd ( L+ 0.8d) iii. The velocity (when n and L and d are constant) d n dv L + 0.8d b. For d = 0.5, nv = 400, use the differential df to approimate the change in frequency when the length of the tube increases from L = 0.9 to L =.0 Recall that Δf df = f '0.9 SOLN: Δ L 400 40 df = 0. = = 0 Hz 0.9 0.8 0.5 ( ) + 0.9 ( + 0.). The chart at right tabulates the populations (in millions) of Gabon and Gambia in the years 960 and 980. Let (t) and (t) represent the populations Gabon and Gambia, respectively. Assume that each population follows an eponential growth model: = 0 e rt, where t = 0 in the year 960. a. Set up equation by evaluating the ratio, ( 0) ( 0), in terms of both the data and the formula. Then solve these to find the relative growth rates r and r for Gabon and Gambia populations, respectively. Simplify what you can without using a calculator (don t appro. the logarithms.) 0r ( 0) e 0 0r 686 ln.4 SOLN: For Gabon, = = e = =.4 r = ( 0) 0 490 0 0r ( 0) e 0 0r 30 ln For Gambia, = = e = = r = ( 0) 0 60 0 b. What equation would you solve to predict, based on this evidence, the time when the populations of Gabon and Gambia are equal? Don t solve the equation here, maybe later today. ln.4 t/0 ln t/0 t SOLN: 490e = 30e 490(.4) = 30( ) /0 t /0
3. A camera is positioned at the top of a tower of height 0.3 km. The base of the tower is km from the base of a rocket launching pad, as shown in the diagram. Assume the height (in km) of the rocket at time t (in sec) is given by () t = 0.003t Let θ be the angle of elevation as shown in the diagram at right. a. Find an equation relating and θ in terms of t. 0.3 SOLN: tanθ = b. Find the rate of change of θ when θ = 0. SOLN: Find the rate of change of θ per change in when θ = 0. d d tanθ = ( 0.3 ) sec θ = = cos θ, so at θ = 0, = Now this will occur when t = 0, at which time = = ( 0.06) = 0.06 rad/sec dt dt = 0.006t = 0.06 so that t= 0 dt 4. The gas law for a fied mass m (in moles) of an ideal gas at absolute temperature T (in Kelvins), pressure (in atmospheres), and volume V (in liters) is V = mrt, where R 0.8 is the gas constant. Suppose that, at some instant, = 0 atm and is decreasing at a rate of 0. atm/sec, the volume is V = 8 L and V is increasing at a rate of 0.5 L/sec. Find the rate of change of the temperature at that instant if there are 00 moles of gas. d d SOLN: V = mrt ' V + V ' = mrt '. Now = 0, = 0., V = 8 and V = 0.5, so dt dt.4 57 0.( 8) + 0( 0.5) = 00( 0.8 ) T' 6.4 + 5 = 8 T' T' = = degrees K per sec. 8 40 5. Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a sphere with diameter 6 cm. Hint: The paint increases the volume of sphere slightly, and the formula for the 4 3 volume V of a sphere in terms of its radius r is V = π r. 3 dv SOLN: The volume of paint is 9π ΔV dv = Δ r = 4π 3 ( 0.05) =.8π = cm 3. dr 5 6. Find a formula for the derivative function for y sinh d = ( ) Recall that sinh =. + SOLN: y' = sinh ( ) + + 4
7. At what point along the curve y = cosh() does the tangent line have slope? e e y' = sinh( ) = = e e = e e = ( e ) = SOLN: e = = ln +